The Existence Of Solutions For Several Classes Of Elliptic Equations With Variable Exponent

The study of variable exponent function spaces has received considerable attentionin recent years. With the development of the field of elastic mechanics and electrorheo-logicalΩuids etc, variable exponent function spaces demonstrate the practicability. Forexample, many researchers deal with the functional integral with the p(x)-growth condi-tions during the study of the problems in elastic mechanics. variable exponent functionspaces provide the theory frame for the study of this problem. The study of diΩerentialequation with p(x)-growth conditions and the variational problem associated with thisequation has become the new topic.In this paper, we study a class of p(x)-Laplacian equations with singular term,and higher order quasilinear elliptic equation, based on the theory of variable exponentLebesgue space Lp(x)(Ω) and Sobolev space Wm, p(x)(Ω), whereΩΩRN. We discuss theenergy functional associated with the p(x)-Laplacian equation with singular term. For thehigher order quasilinear elliptic equation, we construct a sequence of solutions, and thenwe discuss the limit. As the exponent p(x) is a function onΩ, there exists many newproblems during the process of the study. We use new techniques to obtain the results weneed.In this paper, we mainly discuss the following contents:1. We discuss the existence and multiplicity of weak solutions for a class of p(x)-Laplacian equations with singular term. We study the critical points of the energy func-tional I associated with the p(x)-Laplacian equation. As the change of the growth expo-nent, we use the mountain pass theorem to get the nontrivial critical point u∈W01 , p(x)(Ω)of I, i.e. we get the existence of nontrivial weak solution for the equation. Then by theFountain theorem, we obtain a sequence of critical points {un}ΩW01 , p(x)(Ω) of I, andI(un)→∞, i.e. we obtain the multiplicity of weak solutions for the equation. ApplyingEkeland's variational principle, we also get the nontrivial critical point of u∈W01 , p(x)(Ω).2. We discuss the multiplicity of weak solutions for a class of p(x)-Laplacian e-quations with the critical Sobolev-Hardy exponent. According to the principle of con-centration compactness in variable Sobolev space W01 , p(x)(Ω), we establish a principle ofconcentration compactness that we need. Furthermore, by the Fountain theorem, we ob-tain a sequence of critical points {un}ΩW01 , p(x)(Ω) of I, and I(un)→∞. Thus we get the multiplicity of weak solutions for the equation.3. We discuss a class of the Dirichlet problem for higher order quasilinear ellipticequation. We first construct the finite dimension subspace S JΩW0m , p(x)(Ω), where Jis a positive integer. For every fixed J, we get the existence of the weak solution forthe equation in S J. As the change of J, we obtain a sequence of weak solutions {uJ}.Furthermore, by the property of the S J space, we have un→u in W0m , p(x)(Ω), i.e. u is thenontrivial weak solution for the equation that we study in W0m , p(x)(Ω).4. We discuss a class of the Neumann boundary value problem for higher orderquasilinear elliptic equation. In Wm, p(x)(Ω) space, we first construct a class of equation,for which we obtain the existence of the weak solutions {un} for every fixed n. By theproperty of {un}, we have un→u in Wm, p(x)(Ω), i.e. u is the nontrivial weak solution forthe equation that we study in Wm, p(x)(Ω).In this paper, the research about the equation with p(x)-growth conditions is moreuseful than that with p-growth conditions in practice.